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Edited By Kuldeep Maurya | Updated on Jan 21, 2022 01:03 PM IST

Many students widely use the RD Sharma solution books to clarify their doubts without the help of a teacher or a tutor. Solving the Very Short Answers (VSA) in class 12 mathematics is trickier as the solutions must be small and solved using shortcut methods. However, students can save time and complete the sums effortlessly using this method. And to learn it, the RD Sharma Class 12th VSA book is the best choice.

Chapter 15 - Tangents and Normals Ex 15.1

Chapter 15 - Tangents and Normals Ex 15.2

Chapter 15 - Tangents and Normals Ex 15.3

Chapter 15 -Tangents and Normals Ex-FBQ

Chapter 15 -Tangents and Normals Ex-MCQ

Tangents and Normals Exercise Very Short Answer Question 1

Answer :Hint:

Use the slope of the tangent.

Given:

Here the curve , where the tangent is parallel to axis.

We have to find the point on the given curve.

Solution :

The slope at the axis is .

Now,

Let be the required point.

Since the point lies on the curve.

Hence ....(i)

Then,

Slope of the tangent at

Here

From equation (i)

Hence required point is

Tangents and Normals Exercise Very Short Answer Question 2

Answer : Slope of tangent atHint : Use equation of tangent,

Given :

Here,

We have to find the slope of tangent to given curve at

Solution :

The equation of the given curve is

Differentiating equation (i) with respect to

We have,

Now, slope of tangent at

Hence, Slope of tangent at is .

Tangents and Normals Exercise Very Short Answer Question 3

Answer :Hint : The curve is parallel to axis then slope of tangent is equal to slope of the axis.

Given :

The tangent line at a point on the curve is parallel to axis.

We have to find the value of .

Solution :

Here we have,

Slope of axis is equal to zero.

Also, the tangent line at a point on the curve is parallel to axis.

Slope of tangent slope of axis

Tangents and Normals Exercise Very Short Answer Question 4

Answer :Hint :

1. The slope of y - axis is .

2. Using,

Given: The normal of the curve at is parallel to y -axis.

We have to write the value of

Solution:

We know that,

The slope of y - axis is

Also, The normal of the curve is parallel to y - axis.

slope of normal = slope of y- axis

Hence,

Tangents and Normals Exercise Very Short Answer Question 5

Answer :Hint:

The curve is equally inclined to the axes, then the angle made by the tangent with axes can be .

Given:

Here given that,

The tangent to a curve at a point is equally inclined to the co-ordinate axes.

We have to write the value of .

Solution :

We know that,

The tangent to a curve at a point is equally inclined to the co-ordinate axes, the angle made by the tangent with axes can be .

Hence, or

Tangents and Normals Exercise Very Short Answer Question 6

Answer :Hint : Slope of the y-axis is .

Given :

Given that

The tangent line at a point on the curvey is parallel to y-axis.

We have to write the value of .

Solution :

We know that,

The slope of the y-axis is .

Also, the tangent line at a point on the curvey is parallel to y-axis.

= slope of y-axis

Hence,

Tangents and Normals Exercise Very Short Answer Question 7

Answer :Hint : Usingmula,

Given :

Here given the curve,

We have to find the slope of normal.

Solution :

Here,

Differentiating with respect to , we get

So, Slope of tangent

Hence,

Tangents and Normals Exercise Very Short Answer Question 8

Answer :Hint :

Given :

Here the given curve, where the tangent line makes an angle with x- axis.

We have to find the co-ordinate of the point on the given curve.

Solution:

Let the required point be .

We know,

The tangent makes an angle with the x-axis.

Since the point lies on the curve

Hence,

Now,

Differentiating with respect to x,

Now, we have,

Hence,

Tangents and Normals Exercise Very Short Answer Question 9

Answer :Hint : Differentiating both equation and find

Given :

Given that the curve,

We have to find the angle made by the tangent to the given curve.

Solution :

Here,

Differentiating both equation with respect to

We have,

Now, slope of tangent

Let be the angle made by the tangent with the x-axis.

Hence, the angle made by the tangent to the given curve is .

Tangents and Normals Exercise Very Short Answer Question 10

Answer : Equation of normalHint : Use equation of normal,

Given :

Here given that the curve

We have to write the equation of normal to the given curve at

Solution:

We have,

On differentiating both sides with respect to , we get

Now we know that,

Now,

When

So, equation of normal

Hence the equation of the given curve at .

Tangents and Normals Exercise Very short Answers Question 11

Answer :Hint :

For line, the slope of a line is denoted by m.

, where m is slope of line.

Given :

Given that the curve,

Where the tangent parallel to the line,

We have to find the co-ordinate of the point on the given curve.

Solution :

Let be the required point

We know that,

If

Comparing with equation:

We get,

Slope of the given line

Since, the point lies on the curve

So ...(i)

Now,

Differentiate with respect to x

Slope of tangent

Here, slope of tangent=slope of the line

From equation (i), we get

Hence,

Tangents and Normals Exercise Very short Answers Question 12

Answer : Equation of tangent,Hint: Use the equation of tangent,

Given : Here the curve,

at the point where it crosses the y-axis.

We have to write the equation of the tangent.

Solution :

We know that,

When the curve passes through y - axis, then the point on the curve is of them

Now,

Differentiating with respect to x,

Slope of tangent

Equation of tangent,

Hence the required equation of tangent, .

Tangents and Normals Exercise Very short Answers Question 13

Answer : Angle =Hint : First, find the slope of given curves then use themula:

Given :

Given curve,

We have to find the angle between the given curves.

Solution :

Given :

...(i)

...(ii)

On differentiating equation (i) with respect to x, we get

Now, differentiating equation (ii) with respect to x, we get

As we know that,

Hence the required angle between the curve is

Tangents and Normals Exercise Very short Answers Question 14

Answer : Required angleHint :

- If , they are parallel.
- If , they are perpendicular to each other.

Given that the curve,

We have to find the angle between the given curves at the point of intersection.

Solution :

Given,

....(i)

.....(ii)

From (i) and (ii), we get

Substituting the value of x in equation (ii), we get

So, the point of intersection of the two curves is

On differentiating (i) with respect to x, we get

Hence, they are perpendicular to each other.

Hence, the required angle

Tangents and Normals Exercise Very short Answers Question 15

Answer : Slope of normal = 9Hint :

Given :

Given curve,

We have to write the slope of normal to the given curve at point

Solution :

Here,

On differentiating with respect to x, we get

Now,

Hence, the Slope of normal = 9

Tangent and Normals Exercise Very short Answers Question 17

Given :

Given curve,

We have to write the slope of normal to the given curve at point

Solution :

Here,

On differentiating with respect to x, we get

Now,

Hence, normal line is x=0.

Tangent and Normals Exercise Very short Answers Question 18

Hint :

Given :

Given curve,

We have to write the equation of tangent drawn to the given curve at point (0,0).

Solution:

Differentiating both sides with respect to x, we get

Hence the equation of tangent at ,

Hence, is the required equation.

Tangent and Normals Exercise Very short Answers Question 18

Hint :

Given :

Given curve,

We have to write the equation of tangent drawn to the given curve at point (0,0).

Solution:

Differentiating both sides with respect to x, we get

Hence the equation of tangent at ,

Hence, is the required equation.

Tangent and Normals Exercise Very short Answers Question 19

Answer :

Hint :

Simply we will find at

Given:

Given curve,

We have to find the slope of the tangent to the given curve at

Solution :

Differentiating both sides with respect to x, we get

Hence the slope of tangent = 0

Chapter 15 in mathematics, Tangents, and Normals consists of three exercises, ex 15.1, ex 15.2, and ex 15.3. Class 12 RD Sharma Chapter 15 VSA Solution includes concepts like finding the point of the curve, finding the slope of the tangent, slope of normal at a point, equation of the normal, coordinates of the point, and angles between the curve. RD Sharma Class 12 Solutions Tangents and Normals VSA are around 19 questions given in this exercise. Students can use the RD Sharma Class 12 Chapter 15 VSA solution book for reference.

It is mainly recommended by the CBSE board schools to its students, as the RD Sharma solutions follow the NCERT syllabus. Students can work out the practice questions to understand the steps clearly. The RD Sharma Class 12th VSA book answers are given in simple, easy methods to solve the Very Short Answers quickly. It helps the student use minimal time to solve this section during examinations.

If you are wasting time working out VSA solutions in an elaborated manner, use the Class 12 RD Sharma Chapter 15 VSA to understand how easily sums can be solved. With good practice, you'll be able to solve the Very Short Answers of the Tangents and Normals chapter in a short period. This helps you save time that can be spent rechecking the answer sheet.

Looking at the benefits provided by the RD Sharma book, you might presume that would cost a lot. But here is a simple solution for you, the RD Sharma Class 12 Solutions VSA book can be downloaded from the Career360 website for free of cost. It saves you from spending hundreds to thousands of rupees to purchase solution books. The top educational website, Career360, also lets you download other solutions books that you require.

There is a high chance the questions for the public exam will be asked from the RD Sharma Class 12th VSA practice questions. You can use this book to compete for your homework, do assignments and even prepare for your tests and exams. Practising with the RD Sharma Class 12 Solutions Chapter 15 VSA will increase your efficiency and speed in the concept. Eventually, you can witness yourself scoring good marks in the exams.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. Which reference material can the students depend on to recheck their answers and clarify doubts in maths chapter 15 VSA?

The students can avail the benefits of the RD Sharma Class 12th VSA solution book by clarifying their doubts regarding the very short answers given in mathematics chapter 15.

2. Where can I learn the tricks and shortcut methods to solve the Chapter 15 VSA quickly?

The shortcut methods and tricks to solve the VSAs in chapter 15 are given in the RD Sharma Class 12th VSA solution book. With the help of it, you can solve the sums quickly.

3. Where can I find the RD Sharma solution books for free of cost?

The Career 360 website allows its visitors to access the RD Sharma book PDFs for free of cost without charging even a penny.

4. Should I visit the Career 360 website each time there is a doubt in solving the sums?

The Career360 website also provides the option to download the reference materials in the form of PDFs. Therefore, you can refer to the PDFs for further clarifications.

5. Are the RD Sharma solution materials accessible by everyone?

The Career 360 allows everyone to view the RD Sharma solution books for the welfare of the students.

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